The Games and Puzzles Journal — Issue 35, September-October 2004

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# Squaring the Square

## A note on the dissection of a square into smaller squares all different except for a single pair of equal non-touching squares (or two such pairs)

by T. H. Willcocks (28 May 2004)

Editor's Note: This article has been delayed in publication mainly because I was hoping to find a better way of presenting the dissection diagrams. Mr Willcocks provided A4-size diagrams prepared for him by Dr J. D. Skinner, but it was not possible to reproduce them adequately on the screen. However, George Bell (19 December 2004) has now been able to provide a scale drawing gif-image of the first dissection (A1) as shown above. The impossibility of showing an accurate diagram is evident from the fact that the square side is 6732 units and the screen width is only 800 pixels. So the above image, which is 600 pixels wide, compresses 11.22 units to a pixel. So squares smaller than this, as occur in B1, B3, C1, C2 below, would be lost. For the diagrams in the text I have used the conventional method of showing the squares as rectangles.

During the past 75 years much work has been done on the problem of dividing a rectangle into smaller squares all of different sizes. When the rectangle itself is a square the problem is known as Squaring the Square and a comprehensive theory was developed by four mathematicians at Cambridge by which they showed a relationship between the elements of dissections and the currents in certain electrical networks. Details may be found in [1] and [2]. A very interesting non-mathematical but authoritative account may be found in [3]. The following notes are based entirely on the electrical theory.

In the diagram of Fig.(1a) ABC represents a network where for simplicity interior wiring has been omitted. DEF represents the map which is the dual of an equivalent map to ABC (interior wiring again omitted). These two networks are joined by the wires BE, BD, DC, CF, and FA. If now a current enters the network at A and leaves at E, the theory tells us that (1) the network gives rise to a squared square and (2) the wires BD and CF carry zero currents.

Lying in my bath one evening with a mental picture of such a network, I realised that the wire XD could equally well be drawn as XB and similarly the wire YB could be drawn as YD. What I had done was to replace two sides of the quadrilateral BYDX with the other two sides. It seemed that the natural thing to do was to replace the diagonal BD with the other diagonal XY and see what resulted. I had no reason to expect any interesting outcome, but on evaluating the new network I discovered that (1) I had a squared square, (2) the currents in BX and DY were numerically equal, and (3) vectors could be assigned as shown in Fig.(1b). Knowledge of (2) and (3) can simplify the calculation of currents in the network.

Experiment showed that a similar procedure could be followed with the wire CF or with both wires with a view to obtaining a single pair of equal squares in two different ways or two pairs of equal squares in one way. I have constructed a considerable number of such squares without finding a failure, but I lack a strict proof justifying the procedure. As with all other methods of this kind it is important to check that there is no ‘unwanted’ imperfection.

An example of three squares constructed in this way showing an interesting arithmetical relationship is given in the accompanying diagrams C1, C2, C3. The sum of the duplicated squares in C1 and C2 (92 + 208) is equal to the sum of the two duplicated pairs in C3 (97 + 203).

This method is not restricted to the use of self-dual maps. A dissection of a rectangle in two different ways but with a corner square in one equal to a corner square in the other will yield two different tripolar networks and one of these can be used with the dual of the other.

In this way we can form four networks, and it is their close relationships with each other which result in some curious arithmetical equalities, such as that noted above. From A and B we have 431 + 187 = 332 + 286.

The writer does not claim that all initial networks give such interesting results. He has been content to show some that he has found.

Diagrams of the results. The duplicated squares are highlighted in yellow, and the square connecting them in green.

A1: 34 squares, side 6732 units.
 3006 1870 1856 651 1205 1136 734 97 554 831 1456 865 685 513 623 374 1385 194 1011 403 110 927 454 231 591 274 634 317 411 110 2286 2270 94 2176

A2: 34 squares, side 6876 units.
 2637 2305 1934 714 1220 332 1422 551 1834 1135 208 506 759 461 1265 416 804 481 609 332 699 436 28 388 360 263 654 526 83 433 2457 2405 391 2014

A3: 35 squares, side 9447 units.
 3561 3130 2756 1002 1754 431 1821 878 2476 1516 250 752 1128 626 1880 682 708 431 960 556 500 1254 177 254 100 77 404 834 831 808 3410 430 77 3057 2980

B1: 34 squares, side 6705 units.
 2592 1681 2432 911 770 202 2230 141 446 183 1285 1307 747 305 385 629 122 507 560 187 1323 1263 22 939 950 187 2043 302 626 11 615 1856 1565 1241

B2: 34 squares, side 6732 units.
 2432 2293 2007 286 1721 842 1737 1870 562 421 141 374 669 327 94 233 235 302 1419 560 558 286 272 1102 751 2430 830 51 634 166 1585 1600 483 1117

B3: 35 squares, side 4669 units.
 1804 1526 1339 187 1152 604 1109 1490 314 302 12 298 318 294 8 286 20 151 187 144 1008 115 36 774 702 695 1375 115 72 468 162 1260 396 1170 864

C1: 34 squares, side 6705 units.
 2817 2149 1739 410 1329 668 1048 843 1555 777 485 288 380 205 562 76 1405 196 92 1368 357 1 195 292 194 389 919 778 291 680 2333 92 2232 2140

C2: 34 squares, side 6876 units.
 2637 1762 2477 875 887 706 1771 1175 644 818 513 362 353 534 359 3 356 305 208 175 1065 470 174 1098 296 1001 61 705 1236 208 2628 592 2420 1828

C3: 35 squares, side 9413 units.
 3509 3306 2598 708 1890 203 1153 1506 1152 2404 1308 354 768 30 358 380 212 203 1920 53 150 168 44 97 1693 414 1096 570 548 1182 1118 3508 97 3005 2908

It is possible to make further speculations. It is obvious that the process is reversible. The method employed results in all cases with a square containing 3 component squares situated as in Fig.(2a). Reversing the process the two squares A are removed and a zero current is introduced.
Fig.2a
 A B A
Fig.2b
 A B A D A

What happens if we remove by this process in a square with a configuration as in Fig.(2b) the two squares A which adjoin square D? The small number of examples I have tested have all yielded squared squares.

The sceptical reader might care to test the inverse method with the following square, Fig.(3). At first sight this does not look like a candidate but it can be made one by treating the square [4] as being 4 squares of side 2 surrounding a zero square. To my surprise I found that this resulted in a square (imperfect of side 313). Is it conceivable that a perfect square could be found in this way?

Fig.(3). 21 squares, side 107
 34 36 37 32 2 21 17 16 21 4 20 9 9 16 4 17 13 41 34 2 32

In conclusion I would like to express my thanks to Dr J. D. Skinner for his help with the diagrams.

References.
[1] The dissection of rectangles into squares R. L. Brooks, C. A. B. Smith, A. H. Stone, W. T. Tutte, (Duke Math J. Vol.7 p.312-340)
[2] Squaring the Square W. T. Tutte (Can. J. Math. Vol.2 p.197-209, 1950).
[3] Squaring the Square W. T. Tutte (Chapter 17 of the 2nd Scientific American Book of Mathematical Puzzles and Diversions).

George Bell cites the following website on squaring the square, though I've not been able to access it properly since its "javascript" does not pass my computer's security settings. www.squaring.net
 Update (2 June 2013): My more recent computers have had no trouble with access to the site. I have corresponded with the author Stuart Anderson. There is now a page devoted to T. H. Willcocks and his work: THW on www.squaring.net. Mr Willcocks reached the age of 100 on 19 April 2012.